Today, chaotic systems have become one of the most important tools for encrypting and secure transmission of information. Other applications of these systems in economics, geography, sociology, and the like are not hidden from anyone. Despite the presentation of various chaotic systems, it is necessary to study and present new and more accurate chaotic systems. It is obvious that fractional models are more accurate and yield better results than integer order models. In this paper, the synchronization and anti-synchronization of an innovative fractional order chaotic system is investigated based on the nonlinear control method. In the proposed chaotic system, there is an exponential term that leads to behaviour very different from the integer order chaotic systems. Two different approaches have been proposed to achieve the synchronization and anti-synchronization goals between the proposed new fractional chaotic systems. A backstopping approach has been used to synchronize, and in addition to achieving this goal, it also ensures stability in Lyapunov's concept. Anti-synchronization between the two new fractional systems is also achieved by applying the active control method, and subsequently Lyapunov stability is shown under the proposed method. The simulation results in MATLAB environment show the synchronization and anti-synchronization effectiveness for the proposed innovative fractional order chaotic system.

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